image magnification using interval information

3112 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 20, NO. 11, NOVEMBER 2011

Image Magnification Using Interval InformationAranzazu Jurio, Miguel Pagola, Radko Mesiar, Gleb Beliakov, Senior Member, IEEE, and

Humberto Bustince, Member, IEEE

Abstract—In this paper, a simple and effective image-magnifi-cation algorithm based on intervals is proposed. A low-resolutionimage is magnified to form a high-resolution image using ablock-expanding method. Our proposed method associates eachpixel with an interval obtained by a weighted aggregation of thepixels in its neighborhood. From the interval and with a linearoperator, we obtain the magnified image. Experimental resultsshow that our algorithm provides a magnified image with betterquality (peak signal-to-noise ratio) than several existing methods.

Index Terms—Image magnification, implication operator, in-terval, operator.

I. INTRODUCTION

I MAGE-MAGNIFICATION methods attempt to increasethe spatial resolution of an image without introducing a

blur. This process is also known as superresolution [1], [2],image scaling [3], upsampling [4], zooming [5], and imageenlargement [6] in related literature.Image magnification is currently a very active area of re-

search as it is used in many applications. For instance, it canovercome resolution limitations of low-cost imaging systems ofpersonal digital assistants, mobile phones, or surveillance cam-eras. It is also considered as key medical- and satellite-imagingtechniques where the diagnosis or analysis in enhanced reso-lution images is desired. Additionally, there exists a great de-mand of video sequence enhancement [7] due to the fact thatmost web videos are limited by network bandwidth and serverstorage; thus, it should be stored in low quality.Previous works on image magnification can be roughly di-

vided into three categories: 1) reconstruction methods; 2) ap-proaches based on machine learning techniques; and 3) interpo-lation methods.

Manuscript received August 24, 2010; revised January 26, 2011 and April 06,2011; accepted May 16, 2011. Date of publication May 31, 2011; date of cur-rent version October 19, 2011. This paper was supported in part by the NationalScience Foundation of Spain under Grant TIN2010-15055, by the Research Ser-vices of the Universidad Publica de Navarra, by Grant VEGA 1/0080/10, andby Grant P402/11/0378. The Associate Editor coordinating the review of thismanuscript and approving it for publication was Prof. A. A. Ross.A. Jurio, M. Pagola, and H. Bustince are with the Departamento de

Automatica y Computacion, Universidad Publica de Navarra, 31006 Pam-plona, Spain (e-mail [email protected]; [email protected];[email protected]).R. Mesiar is with the Department of Mathematics and Descriptive Geom-

etry, Slovak University of Technology, 813 68 Bratislava, Slovakia, with theInstitute of Information Theory and Automation, Academy of Sciences of theCzech Republic, 18208 Prague, Czech Republic, and also with the Institute forResearch and Application of FuzzyModeling, University of Ostrava, 70103 Os-trava, Czech Republic (e-mail: [email protected]).G. Beliakov is with the School Information Technology, Deakin University,

Burwood 3125, Australia (e-mail: [email protected]).Digital Object Identifier 10.1109/TIP.2011.2158227

The reconstruction task is based on prior knowledge aboutthe model that maps the high-resolution image to the low-res-olution one. The magnification is then considered as the in-verse problem: recovering the original high-resolution image byfusing one or more low-resolution images [8].In the approaches based on machine learning methods

[9]–[11], a set of examples of low resolution and their idealhigh-resolution patches are organized in a database. The mainidea consists on seeking in the database for similar low-reso-lution examples, given a patch from the low-resolution input.Their corresponding high-resolution counterparts can then beused for the reconstruction.The most frequently used methods working with a single

image are based on interpolation [12]. Common algorithms,such as nearest neighbor or bilinear interpolation, are computa-tionally simple but suffer from smudge problems, particularly inthe edge areas. Directional image interpolations take advantageof the geometric regularity of image structures by performingthe interpolation in a chosen direction along which the imageis locally regular. Therefore, many adaptive interpolations havebeen developed with edge detectors [4]. Nevertheless, linear ap-proximations are the most used ones since their computationalcost is lower.Our goal in this paper is to develop a cost-efficient method

to construct, starting from a given image, a new image of largersize such that each area or block in the new image is obtainedby a weighted aggregation of the intensities of the pixels in theneighborhood of each pixel in the original image. To reach thisgoal, the method that we propose makes use of the notions ofinterval and operators [13]. We use intervals because it hasbeen proven in image processing that they allow keeping infor-mation about the neighborhood of each pixel [14]. In this way,we employ that information to get the intensities of the newpixels in the magnified image. We also use operators be-cause they allow us to choose, depending on the parameter,the internal point of the interval that we must select.To associate an interval to each pixel within the image, we

present a new construction method of intervals from a pixel andits neighborhood. We are going to understand the length of thisinterval as a measure of the variation of intensities in the neigh-borhood of each pixel.To see that our method is computationally simple and gets

good results, we compare the images obtained by our algorithmand the ones obtained by other methods, both classical and morerecent ones.This paper is organized as follows: First, in Section II, we

recall some preliminary definitions. In Section III, we showthe method to construct intervals. In Section IV, we presentthe image-magnification algorithm. We finish with some exper-imental results in Section V and conclusions in Section VI.

1057-7149/$26.00 © 2011 IEEE

JURIO et al.: IMAGE MAGNIFICATION USING INTERVAL INFORMATION 3113

II. PRELIMINARIES

In this section, we recall the main concepts that are necessaryfor subsequent developments. A strictly decreasing and contin-uous function , such that and

, is called strict negation. If, in addition, is invo-lutive , then we say that it is a strongnegation. We call automorphism of the unit interval every func-tion that is continuous, strictly increasing,and such that and .Let us denote by the set of all closed subintervals in

[0, 1], that is

and

is a lattice with respect to the relation , which isdefined in the following way, given , , we have

if and only if and

From now on, we denote by the length (width) of theconsidered interval; that is, .Definition 1: Let . The operatoris given by

for all .Clearly, the following properties hold.a) for all .b) for all .c)

for all .Notice that operator is a particular case of the Hurwicz

aggregation function [15].

A. Implication Operators

Definition 2: An implication operator is functionthat satisfies the following properties

[16].: implies for all .: implies for all .: for all (dominance of falsity).: for all .: .

Remark: Properties , , and imply that is an exten-sion of the standard Boolean implication operators. Indeed, itholds , and .We follow the notation presented in [16], where other proper-ties can be found. In particular, we will use the following onesthroughout this paper.

: (neutrality of truth).: (exchange property).: where is a strong negation.: is a continuous function (continuity).

III. CONSTRUCTION OF INTERVALS OF FIXED LENGTH

In our algorithm, we represent the information of the neigh-borhood of each pixel by an interval. We use this interval to con-struct the intensities of the new pixels in the magnified image. In

this section, we propose a construction method of the elementsof from two points in [0, 1], such that the first numberis an internal point of the interval and the second number repre-sents the length of the interval.The construction method is described in the following

theorem.Theorem 1: Let be such that(F1) for all ;(F2) is increasing in the first variable;(F3) is decreasing in the second variable;(F4) for all .

Then, , , for all, and mapping

given by

is such that for all , .Proof: By construction, . From

(F4), for all . By (F2), .Thus, is well defined.Due to (F1) and (F3), .

by construction. From (F2),by (F4). As and ,

by (F3).In the following lemma, we relate our operator that gen-

erates intervals with implication operators. This result is reallyuseful because implication operators have been widely studied;thus, we can use several expressions of implication operators tocreate functions.Lemma 1: Let be such that (F1)–(F4)

hold. Then, mapping

given by

is an implication operator for any strict negation .Proof: and follow from (F2) and (F3).

. .from (F4).

In the following theorem, we present a characterization of theconstruction method.Theorem 2: Mapping satisfies properties

(F1)–(F4) if and only if there exists an implication operator sat-isfying with respect to the standard negation and such that

Proof: (Sufficiency) is decreasing in the first variableand increasing in the second, from (F2) and (F3).

. .; thus, .

. (Necessity)from . (F2) and (F3) from the monotonicity properties of .

from .Example 1: In this example, we present some expressions offunctions constructed from different implication operators.a) Consider Kleene–Dienes implication

and standard negation. Then,


. In this way, theresulting interval is

(1)

b) Take Reichenbach implication andstandard negation. Then, .Thus, the resulting interval is

(2)

c) Take Lukasiewicz implication and stan-dard negation. Then,

. Thus, the resulting interval is

(3)

Remark: From now on, we will use the standard negation as.The following theorem is the basis for the image-magnifica-

tion algorithmic construction that we present in Section IV. It isused to keep the intensity of each pixel from the original imagein the new magnified image.Theorem 3: In the setting of Theorem 2, the following item

holds:

if and only if

(Reichenbach's implication).

Proof: (Necessity) .Thus, . (Sufficiency)

.Theorem 4: If withfor all , , then for all .Proof: If , then

. , and .In the next corollary, we build mappings from auto-

morphisms.Corollary 1: Let be such that (F1)–(F4)

hold. Suppose that the implication operator given by Theorem2 satisfies , , and . In these conditions, there exists anautomorphism in the unit interval such that

(4)

Proof: Direct from the result,proved in [16].

Remark: Notice that for (4) to satisfy , it is necessary totake . Therefore, .Example 2: If we take , then

(Reichenbach’s implication).

IV. MAGNIFICATION ALGORITHM

We consider image as an matrix. Each coordinateof the pixels in image is denoted by . The normalizedintensity or normalized gray level of the pixel located atis represented as , with for each .

Fig. 1. How to select grid . For the magnification of the pixel in dark gray,all the pixels in (dark and light) gray belong to grid . (a) The case where themagnification factor is (2 2). (b) The case where the magnification factor isgreater than or equal to (3 3).

Fig. 2. How to select grid for different magnification factors: (a) factor (22), (b) factor (3 3), (c) factor (4 4), and (d) factor .

The purpose of our algorithm is, given image of dimension, to magnify it times; that is, to build a new image

of dimension with , , and, with and . We denote as amagnification factor.To do so, we use two different grids over each pixel in the

original image, in the following way:1) Grid from which we will calculate an interval of inten-sities. This block captures the variability of the intensi-ties in the pixel’s neighborhood. The size of this window is

with , , such that and .The values of and depend on the magnification factor(see Fig. 1).

2) Grid whose size is (see Fig. 2). We use this gridand the interval calculated before to build a new block ofsize in the magnified image.

The scheme of our algorithm is illustrated in Fig. 3. Theimage on the left is the original one. We assign to the pixelthe expanded block of the same size as (in this example,

; hence, is a 3 3 block). The image on theright corresponds to the result of the algorithm.Algorithm 1 presents the method we propose using grids ,, and . In it, we can see that this method is characterized

by its simplicity from the implementation point of view.


Fig. 3. Magnified image with .

Algorithm 1

INPUT: original image, magnification factor.

1. Select , .

2. FOR each pixel DO

2.1 Fix a grid of dimension centered at .

2.2 Calculate as the difference between the largestand the smallest intensities of the pixels in .

2.3 Calculate , where , beingthe standard deviation of the intensities of .

2.4 Build the interval using (2).

2.5 Fix a grid of dimension centered at .

2.6 Build a new empty block of size .

2.7 FOR each element of DO

ENDFOR

2.8 Put the block in the magnified image.

ENDFOR

Next, we explain the steps of Algorithm 1 by means ofan example. Given an image in Fig. 4 of dimension 5 5,we want to build a magnified image of dimension 15 15magnification factor .

A. Step 1: Select ,

These parameters represent the size of grid . Their values( for rows and for columns) are selected depending on thefollowing magnification factor:

if thenelse if thenelse

Fig. 4. Example: Original image.

Fig. 5. Example: Original image.

which is analogous for and . We just use the set of valuesbecause the greater the size of the grid , the greater

the neighborhood of each pixel that we consider, and we do notwant that remote pixels interfere in the magnification.In the example, we want to magnify the image by factor (3

3); therefore, .

B. Step 2.1: Fix Grid of Dimension Centered at EachPixel

This grid represents the neighborhood that is used to buildthe interval. In the example, for pixel (2, 3) (marked in dark grayin Fig. 5), we fix a grid of dimension around it (inlight gray).Remark: For pixels in the first or the last row/column, we

choose a grid centered at them, as shown in Fig. 6.

C. Step 2.2: Calculate as the Difference Between theLargest and the Smallest of the Intensities of the Pixels in

is the maximum length that the interval we are going tobuild can have, i.e.,

(5)

In the example, for pixel (2, 3), we calculate as

D. Step 2.3: Calculate

represents the length of the interval that we build. Ourfirst idea was to use as the length of the interval (instead of

) in our construction. However, that proposal has some


Fig. 6. Example: Grid for pixels in the first or last row/column. (a) Pixel in thefirst column. (b) Pixel in the last column. (c) Pixel in the first row. (d) Pixel inthe last row.

Fig. 7. Problem with the edges in binary images if the amplitude is .

problems, as it is shown in the example in Fig. 7. In the upperpart, it is the numerical calculation and below is its representa-tion in the image, where 0 means black and 1 means white.As we can see, the edge in the magnified image is not correct.

To solve this problem, the length of the interval is proportionallyreduced to parameter , and in this way, the jump from white toblack is gradual. To calculate the value, we use the standarddeviation of the intensities in grid . We use the expression

because the maximum standarddeviation in grid is 0.5, and we want the value to vary in [0,1]. In this sense, if all the pixels in have the same intensity,the amplitude of the interval is ; nevertheless, whenthere is a big difference between the intensities (presence of anedge), the value of decreases, and therefore, the amplitude alsodecreases. Fig. 8 shows the effect of parameter .

Fig. 8. Solution to the problem in Fig. 7 using parameter.

Fig. 9. Original block for pixel (2, 3).

Following the example in Fig. 4, the standard deviation ofgrid is 0.0834; thus, .Then, the length for the interval associated to pixel (2, 3) is

.

E. Step 2.4: Build Interval

We associate to each pixel an interval of length . To doso, we use the method explained in Section III. In this case, wetake the expression in (2) based on Reichenbach’s implication.The resulting interval is

In the example, the interval associated to pixel (2, 3) is given by

F. Step 2.5: Fix Grid of Dimension Centered at

In the example, this new block is shown in Fig. 9.

G. Step 2.6: Build a New Empty Block of Size

Following the example, the size of this block is 3 3 (seeFig. 10).


Fig. 10. Empty block .

Fig. 11. Expanded block for pixel .

Fig. 12. Numerical expanded block for pixel .

H. Step 2.7: Calculate for Each Pixel

Next, we expand pixel in image over the new block. In the example, pixel (2, 3) is expanded, as shown in Fig. 11.To keep the value of the original pixel at the center of the new

block, Theorem 3 states that should be equal to the intensityof that pixel. In the case of pixel (2, 3), we have

We apply this method to fill in all the other pixels in the block.In this way, from Proposition 3, we take as for each pixel thevalue of that pixel in the grid .• . Then,

.• . Then,

.•• . Then,

.In Fig. 12, we show the expanded block for pixel (2, 3) in the

example.Once each of the pixels has been expanded, we join all the

blocks (Step 2.8) to create the magnified image. This process isshown in Fig. 3.

V. EXPERIMENTAL RESULTS

To quantitatively estimate the quality of magnification, wetake a set of gray-scale images of size 512 512 and reduce

Fig. 13. Schema to check the algorithm.

Fig. 14. Images used in the different factors experiment. (From left to right)Ship, Lena, and Peppers.

TABLE IPSNRS (IN DECIBELS) OVER IMAGES IN FIG. 14. (FROM LEFT TO

RIGHT) IMAGES MAGNIFIED BY FACTORS (1 2),(2 1), (2 2), (3 3), AND (4 4)

them to several sizes (see Section V-A). Then, we enlarge thepreviously reduced images back to 512 512 and comparethem to the images from which we started. Our expectation isthat the enlarged images will be similar to the original ones,and we can estimate the similarity quantitatively by the peaksignal-to-noise ratio (PSNR).In Fig. 13, we see a schema of the procedure when we reduce

the image to obtain a 9 smaller one and then enlarge the resultby the magnification factor (3 3).

A. Reduction Algorithm

To reduce the images, we use the algorithm proposed in [17].Starting from the images of dimension , the scheme ofthe algorithm is the following.1) Divide image in blocks of size . If is not a multipleof or of , we delete the minimum number of rows/columns in the boundary of the image until the new size ofthe image satisfies the property.

2) Associate each block with an interval in the following way:The lower bound of the interval is given by the minimumof the intensities in the block and the upper bound, by themaximum.

3) Associate each interval with the middle point of theinterval.

We select this reduction algorithm because it takes intoaccount all the pixels of the block, instead of other reductionmethods such as subsampling (take one of every pixels).


Fig. 15. (a) Reduced image Ship of size 256 256 and its magnification by (b) factor (1 2) of size 256 512, (c) factor (2 1) of size 512 256, and(d) factor (2 2) of size . (e) Reduced image Ship of size 170 170 and (f) its magnification by factor (3 3) of size 510 510. (g) Reduced imageShip of size 128 128 and (h) its magnification by factor (4 4) of size 512 512.

Fig. 16. (a) Reduced image Lena of size 256 256 and its magnification by (b) factor (1 2) of size 256 512, (c) factor (2 1) of size 512 256, and(d) factor (2 2) of size 512 512. (e) Reduced image Lena of size 170 170 and (f) its magnification by factor (3 3) of size 510 510. (g) Reduced imageLena of size 128 128 and (h) its magnification by factor (4 4) of size 512 512.


Fig. 17. (a) Reduced image Peppers of size 256 256 and its magnification by (b) factor (1 2) of size 256 512, (c) factor (2 1) of size 512 256 and(d) factor (2 2) of size 512 512. (e) Reduced image Peppers of size 170 170 and (f) its magnification by factor (3 3) of size 510 510. (g) Reducedimage Peppers of size 128 128 and (h) its magnification by factor (4 4) of size 512 512.

TABLE IIPSNRS (IN DECIBELS) OVER IMAGES SHIP, LENA, AND PEPPERS AND THE AVERAGE PSNR

OVER 96 IMAGES, WHEN THE MAGNIFICATION FACTOR IS (2 2)

TABLE IIIPSNRS (IN DECIBELS) OVER IMAGES SHIP, LENA, AND PEPPERS AND THE AVERAGE PSNR

OVER 96 IMAGES, WHEN THE MAGNIFICATION FACTOR IS (3 3)

TABLE IVAVERAGE TIME (IN SECONDS) FOR THE MAGNIFICATION OF ONE IMAGE FOR MAGNIFICATION FACTORS (2 2) AND (3 3)


Fig. 18. Cropped images from Ship, Lena, and Peppers. (From top to bottom and from left to right) High-resolution image, magnification by a factor of 2 withnearest neighbor, bilinear, bicubic, and splines interpolations; SME; KR; and Algorithm 1.

B. Results Obtained for Different Magnification Factors inAlgorithm 1

In this experiment, we compare different solutions ob-tained when the magnification factor in Algorithm 1 varies.We take three images from the database [18] (see Fig. 14).In Figs. 15 –17, we show the magnified images using themagnification factors (1 2), (2 1), (2 2), (3 3),and (4 4), as well as the reduced images from which westart in each case.

In Table I, we present the PSNR obtained by each one of theimages. From this table, we see that the obtained results arevery good. Evidently, as it happens with all the magnificationmethods, when the magnification factor is increased, the resultloses quality.

C. Comparison With Other Methods

In this section, we compare our image-magnification algo-rithm with other methods that can be found in the literature.


Fig. 19. Cropped images from Ship, Lena, and Peppers. (From top to bottom and from left to right) High-resolution image, magnification by a factor of 3 withnearest neighbor, bilinear, bicubic, and splines interpolations; KR; and Algorithm 1.

Specifically, we use four classic interpolations [12]: 1) nearestneighbor; 2) bilinear; 3) bicubic; and 4) splines interpolations,as well as three recent superresolution algorithms: 1) sparsemixing estimation (SME) [8]; 2) kernel regression (KR) [19];and 3) sparse representation (SP) [11]. All experiments are per-formed with softwares provided by the authors (SME, KR, andSP) and MATLAB implementation of different interpolations.1

1This paper has supplementary downloadable material available at http://iee-explore.ieee.org, provided by the authors. This includes the MATLAB imple-mentation of Algorithm 1, three example images, and a read-me file. This ma-terial is 104 kB in size.

In the experiment, we work with 96 images from the databasegiven in [18]. The original dimension of all the images is 512512. We reduce them with the reduction algorithm, as explainedin Section V-A, to dimensions 256 256 (using block size 22) and 170 170 (using block size 3 3). We magnify themwith the seven mentioned methods, as well as our method. Wecalculate the similarity between each of the resulting images andthe original images by means of PSNR.Next, we show the different magnifications of the cropped

images Ship, Lena, and Peppers. In Fig. 18, the starting point is


a reduced image of dimension 256 256, and the magnificationprocess has a factor of (2 2). In Fig. 19, the starting point isa reduced image of dimension 170 170, and the magnifica-tion factor is 3 3. In this case, we do not show the SME andSP algorithms because these methods can magnify images by afactor of (2 2).We visually check that the SME, KR, and SP methods get

the results with less jaggy artifacts. However, if we study thePSNR obtained by each method, we check that the quality ofimages magnified with Algorithm 1 is better than the qualityobtained by the rest of the implemented methods, except withthe SP method (see Tables II and III). This happens with thethree shown images and with the average of the 96 images ofthe experiment. In this way, we argue that our proposal worksvery well, and therefore achieves very good results, in all thenonedge areas, i.e., areas where the changes of intensities arenot very big.We have already said that simplicity is one of the strongest

points of Algorithm 1. This fact has an effect on the executiontime of the algorithm. All the experimentation in this paper havebeen executed with a processor Intel Core 2 Duo 6750 at 2.66GHz with 3 GB of random access memory. The software forthe SME and KR methods has been provided by their authors,and the interpolation methods have been implemented by thefunction interp2 of MATLAB, over the version MATLAB 7.8.0(R2009a). In Table IV, the mean time for the magnification ofone image with each one of the implemented methods (for mag-nification factors 2 2 and 3 3) is shown. In this sense, thefastest algorithms are the four classic interpolations, whose timeis always smaller than 0.3 s. However, comparing with the morerecent methods, Algorithm 1 is between 11 and 23 times fasterthan KR, 208 times faster than SME, and 427 times faster thanSP.

VI. CONCLUSION AND FUTURE RESEARCH

In this paper, we have presented an image-magnification al-gorithm that uses intervals and operators. In this method, anew block is constructed for every pixel of the image, and thecentral pixel of that block maintains the intensity of the originalpixel. To fill in the rest of the pixels, we use the relation betweenthe pixel in the original image and its neighbors.In this sense, the algorithm shows that intervals are a good

representation of the effect of the neighborhood over an ele-ment. Due to this, our algorithm provides better results thansome of the most commonly used methods as we have con-firmed with the PSNR values of the experiments. We have alsoproven the computational simplicity of our algorithm.From the results we have obtained, we plan as future research

the study of new methods to calculate the parameter and theuse of filters to improve our results in edge zones.

ACKNOWLEDGMENT

The authors would like to thank the editor and the anony-mous reviewers for their valuable suggestions that have greatlyimproved this paper.

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Aranzazu Jurio received the M.Sc. degree in com-puter sciences in 2008 from the Public Universityof Navarra, Pamplona, Spain, where she is currentlyworking toward the Ph.D. degree and holds a re-search position in the Department of Automatics andComputation.Her research interests are image processing,

focusing on magnification and segmentation, andapplications of fuzzy sets and their extensions.


Miguel Pagola received the M.Sc. degree in in-dustrial engineering from the Public University ofNavarra, Pamplona, Spain, in 2000.He is an Associate Lecturer with the Department

of Automatics and Computation, Public Universityof Navarra. He is the author of over 14 publishedoriginal articles and involved in teaching artificial in-telligence to students of computer science. His re-search interests include fuzzy techniques for imageprocessing, fuzzy set theory, medical image segmen-tation, and medical data mining.

Radko Mesiar received the Ph.D. degree fromComenius University, Bratislava, Slovakia, in 1979and the D.Sc. degree from Czech Academy ofSciences, Prague, Czech Republic, in 1996.He is currently the Head of the Department of

Mathematics, Faculty of Civil Engineering, SlovakUniversity of Technology, Bratislava, Slovakia.He has been a Fellow Member of the Institute ofInformation Theory and Automation, Academy ofSciences of the Czech Republic, Prague, CzechRepublic, since 1995 and the Institute for Research

and Application of Fuzzy Modeling, University of Ostrava, Ostrava, CzechRepublic, since 2006. He is the author of more than 200 papers in Web ofScience and the coauthor of two scientific monographs and five edited volumes.His research interests include the area of uncertainty modeling, fuzzy logic andseveral types of aggregation techniques, nonadditive measures, and integraltheory.Dr. Mesiar is the founder and organizer of the Conferences of Fuzzy Set

Theory and Applications and Aggregation Operators.

Gleb Beliakov (SM’08) received the Ph.D. degree inphysics and mathematics from the Russian PeoplesFriendship University, Moscow, Russia, in 1992.He was a Lecturer and a Research Fellow with Los

Andes University, Bogota, Colombia; the Universityof Melbourne, Victoria, Australia; and the Universityof South Australia, Adelaide, Australia. He is cur-rently an Associate Professor with the School of In-formation Technology, Deakin University, Burwood,Australia. He is the author of a hundred research pa-pers in the areas below and a number of software

packages. His research interests are fuzzy systems, aggregation operators, mul-tivariate approximation, global optimization, decision support systems, and ap-plications of fuzzy systems in health care.

Humberto Bustince (M’06) received the Ph.D. de-gree in mathematics from the Public University ofNavarra, Pamplona, Spain, from 1994.He is a Full Professor with the Department of

Automatics and Computation, Public University ofNavarra. He is the author of more than 80 papers inWeb of Science. His research interests are fuzzy logictheory, extensions of fuzzy sets (type-2 fuzzy sets,interval-valued fuzzy sets, Atanassov’s intuitionisticfuzzy sets), fuzzy measures, aggregation operators,and fuzzy techniques for image processing.

image magnification using interval information

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